presto means soon, fast, quickly, early (早く - まもなく)
subito means immediately, quickly (すぐ)
examples:
il treno arriverà presto
the train will arrive soon
questa mattina mi sono svegliato presto
this morning I woke up early
fai presto!
do it quickly
lo voglio subito!
I want it right now
torno subito
I will be back soon
When you mean "quickly" you can use both:
faccio presto / faccio subito
I do it quickly (in a short while)
Be careful
because "subito" is also the past participle of "subire" (suffer, endure, undergo) sometimes we use the accent to make difference between subito adverb and subito past participle:
sùbito -> adverb
subìto -> past participle
ho subito un torto (past participle of subire)
不当な扱いを受けました
(I hope that I translated in the right way ^^)